A method of simultaneously estimating snow depth and sea
ice thickness using satellite-based freeboard measurements over the Arctic
Ocean during winter was proposed. The ratio of snow depth to ice thickness
(referred to as

Satellite altimeters have been used to estimate sea ice thickness for nearly 2 decades (Laxon et al., 2003, 2013; Kwok et al., 2009). The altimeters do not measure sea ice thickness directly but measure the sea ice freeboard which is then converted to sea ice thickness with assumptions, for example, regarding the snow depth, snow–ice densities, and radar penetration (Ricker et al., 2014). We hereafter refer to this procedure as “freeboard to thickness conversion”.

Generally, there are two types of satellite altimeters measuring different
sea ice freeboards. (1) Lidar altimeters such as NASA's ICESat (Zwally et
al., 2002) and ICESat-2 (Markus et al., 2017) missions measure the total
freeboard (

Schematic diagram of a typical snow–ice system during the winter.
Snow depth (

For both lidar and radar altimeters, snow depth (

However, the use of MW99 for the freeboard to thickness conversion understandably yields a substantial error, considering that W99 is climatology and not actual snow depth. This is because the actual snow depth distribution is subject to the year-to-year variation of the snow–ice system; thus, the climatology based on the 37-year measurements of snow depth would deviate significantly from the actual distribution (Webster et al., 2014). Accordingly, such deviation causes errors in the estimation of ice thickness. Thus, additional snow observations covering both MYI and FYI at the Arctic basin scale would be ideal as a replacement of MW99.

There have been various approaches aimed at obtaining the snow depth distribution over the Arctic scale using satellite observations. Markus and Cavalieri (1998) developed an algorithm based on the brightness temperatures (TBs) of the Special Sensor Microwave/Imager (SSM/I) based on the negative correlation of the snow depth with the spectral gradient ratio between 18 and 37 GHz of vertically polarized TBs on the Antarctic FYI. Comiso et al. (2003) have updated the coefficients of this algorithm for the Advanced Microwave Scanning Radiometer for the Earth Observing System (AMSR-E). However, snow depth retrieval using this algorithm is relatively less accurate when the MYI fraction within the grid cell is significant (Brucker and Markus, 2013). Recently, Rostosky et al. (2018) suggested a new method: using the lower frequency pair of 7 and 19 GHz to overcome this limitation. Nonetheless, estimating the basin-scale snow depth distribution seems to be a difficult task.

There are other approaches involving the use of the lower frequency measurements at L-band. Using Soil Moisture Ocean Salinity (SMOS) measurements, Maaß et al. (2013) found that 1.4 GHz TB depends on the snow depth through the insulation effect of the snow layer, and they determined snow depth by matching TBs simulated with a radiative transfer model (RTM) with SMOS-measured TBs. Zhou et al. (2018) simultaneously estimated the sea ice thickness and snow depth by adding additional laser altimeter freeboard information, improving the Maaß et al. (2013) approach. However, both of these RTM-based approaches require a priori information on ice properties (e.g., temperature and salinity profiles).

Other satellite remote sensing approaches include the snow depth retrieval using dual-frequency altimetry (Guerreiro et al., 2016; Lawrence et al., 2018; Kwok and Markus, 2018), multilinear regression (Kilic et al., 2019), and a neural network approach (Braakmann-Folgmann and Donlon, 2019). In spite of promising results, the dual frequency altimetry method is available only for regions where two altimeters overlap with each other, reducing a great deal of spatial coverage. On the other hand, the regression/neural network methods based on AMSR-2 TBs are prone to the overfitting problem, limiting their applications to other microwave sensors.

Here, let us switch our point of view to solving the buoyancy equation instead of retrieving snow depth directly. Remember that there are two unknowns (snow depth and ice thickness) in the buoyancy equation for given snow–ice densities, freeboard, and assumptions on radar penetration of the snow layer. The attempt so far has been to add one constraint (snow depth information) to the buoyancy equation for solving ice thickness. However, if a particular relationship between two unknowns is available, it can be used to constrain the equation yielding both ice thickness and snow depth simultaneously.

To identify such a relationship, this study examines the vertical thermal
structure within the snow–ice layers observed by drifting buoys. The
vertical thermal structure of a snow–ice system in winter is rather simple;
the temperature profile of the snow–ice system can be assumed to be
piecewise linear, as illustrated in Fig. 1. Therefore, the temperatures at
three interfaces can represent the thermal state of the snow–ice system
fairly well: they are (1) air–snow interface temperature (

Based on this thermal structure, there is a constraint relating the snow depth and ice thickness. In identifying this constraint, conductive heat flux is assumed to be continuous through the snow–ice interface (Maykut and Untersteiner, 1971), implying that conductive heat fluxes within the snow and ice layers are the same under the steady state assumed in the given thermal structure. As the conductive heat flux is proportional to the bulk temperature difference of the layer divided by its thickness, it is possible to deduce the relationship between snow depth and ice thickness from the given thermal structure.

Once the relationship is obtained, then it is possible to apply it at the
Arctic Ocean basin scale because the thermal structure can be resolved from
satellites, as shown in the recently available basin-scale and long-term
satellite-derived interface temperatures (Dybkjær et al., 2020; Lee et
al., 2018). In determining the snow depth along with the ice thickness,
instead of using the snow depth as an input to solve for the ice thickness,
we intend to (1) examine the relationship between the vertical thermal
structure of a snow–ice system (

Here we provide the theoretical background of how the snow–ice thickness
ratio (

We intend to find a relationship between snow depth and ice thickness in
terms of the vertical thermal structure of the snow–ice system. Because the
temperature gradients within the snow and ice layers are linked to both
temperature and thickness, we focus on the temperature gradient. Owing to
the physical reasoning that the conductive heat flux is continuous across
the snow–ice interface (Maykut and Untersteiner, 1971), the following
relationship is valid at the snow–ice interface:

To obtain

The buoy-measured temperature profiles at the vertical resolution of 10 cm
are used in this study (Sect. 3.1). Although the instrument initially sets
the zero-depth reference position to be approximately at the snow–ice
interface, the reference position can deviate from the initial location if
the ice deforms or if the snow refreezes after the temporary melt into
snow ice. In addition, the interfaces (air–snow, snow–ice, and ice–water)
may be located in between measurement levels in a 10 cm spacing. Therefore,
an interface searching algorithm is developed to determine three interfaces
(

The flow chart of the interface searching algorithm:

The interface searching algorithm iterates three processes to find the
location and temperature of each interface: it (1) divides the temperature
profile into four layers using the most recently available locations of the
three interfaces, (2) finds a linear regression line of the temperature
profile at each layer, and (3) updates the location and temperature of each
interface by finding an intersection between two adjacent regression lines.
The algorithm fails if the temperature profile is far from linear or if the
thickness of a certain layer is too thin to have fewer than two data points.
More detailed procedures for determining the interface are provided in Fig. 2 as a flow chart. The outputs are

Examples of interface searching results with an averaging period
of 15 d:

Since

In this section, we describe how

In order to obtain

Before the Arctic-basin-scale retrieval, ice thickness is estimated from OIB
total freeboard measurements using Eq. (11) and from OIB-derived radar
freeboards (Sect. 3.3) using Eq. (12) and using satellite-derived

Here we provide detailed information on the datasets used for the development of the retrieval algorithm, evaluation, and application at the Arctic Ocean basin scale.

To determine the empirical relationship between

Information on the measurement sites of buoys whose observations were used in this study.

Time averages of temperature profiles are used as input for the interface searching algorithm (described in Sect. 2.2) to meet the required near-equilibrium states (e.g., linear temperature profile). However, because of the possibility that the results are dependent on the averaging period, we examine the results using various averaging periods from 1 to 30 d.

For applying the buoy-based

In this study, OIB snow depth (

The 5 years of OIB data during the 2011–2015 period are utilized in this study. The level 4 dataset (Kurtz et al., 2015) during the 2011–2013 period and Quick Look dataset during the 2014–2015 period are obtained from the NSIDC website (see the Data Availability section). Because we use the November–March period for the buoy analysis, only March OIB data are considered for the evaluation. The OIB data are also reformatted into the 25 km grid format by averaging pixel-level OIB observations on the 25 km grid.

For examining the Arctic Ocean basin distribution of ice thickness and snow
depth, CS2 freeboard measurement summary data are used (Kurtz and Harbeck,
2017). They are monthly mean composites of CS2 ice freeboard data in the 25 km Polar Stereographic SSM/I grid format covering the entire Arctic and
available from September 2010. Detailed descriptions of the retracker
algorithm used in this dataset are found in the study by Kurtz et al. (2014). The dataset also includes MW99 (

The CS2 ice freeboard data (

Calculation of

We examine variables (i.e.,

Taking such a two-slope pattern with

Coefficients of the regression equation for averaging periods of 1,
7, 15, and 30 d;

Although the slope pattern discussed with Eq. (15) and Fig. 4 is based on
the weekly averages, the slope pattern seems to be consistent among the data
averaging periods except for an averaging period shorter than 5 d.
Regressions in the form of Eq. (15) are performed with buoy data averaged
with different averaging periods to understand the slope pattern. The regression
coefficients and transition points for the chosen averaging periods are
examined, and results for four averaging periods are given in Table 2.
Detailed information on the coefficients and associated statistics varying
with the averaging period is given in Fig. 5. The positions of slope change
(

According to the regression results, it is possible to estimate

To calculate

We simultaneously retrieved

Simultaneously retrieved ice thickness and snow depth from OIB total and radar freeboards in March of the 2011–2015 period. Corresponding OIB products are at the bottom.

To compare the results quantitatively, scatterplots comparing retrievals
against OIB measurements are made, along with statistics for the snow depth
and ice thickness retrievals, in the top four panels of Fig. 7. The two top-left panels are derived from the use of OIB total freeboard
(

Scatter plots between OIB products and the simultaneously retrieved snow depth and ice thickness from OIB total and radar freeboards during the March 2011–2015 period. Corresponding ice thicknesses estimated from MW99 are in the third row. The red lines are linear regression lines.

Following Eq. (12), snow depth and ice thickness retrievals are made from
the use of radar freeboard measurements, and results are presented in the
two top-right panels in Fig. 7. On the one hand, the comparison of obtained
ice thickness against the OIB ice thickness indicates that the retrieved ice
thickness shows a similar quality as that retrieved from the total freeboard
measurements. On the other hand, snow retrievals from the radar freeboard
show more scattered features compared to snow retrieval results from the
total freeboard. The more scattered features found in the snow depth from the
radar freeboard are likely due to the greater sensitivity of the retrieved

We now examine how the use of MW99 for retrieving sea ice thickness from
ICESat and CS2 measurements compares with results from our simultaneous
method. To do so, OIB-measured total freeboard and radar freeboard are
converted into ice thickness using MW99 as the input to solve Eqs. (4) and (10).
In this study, these two ice thickness retrievals with the use of MW99 are
referred to as “ICESat-like” thickness and “CS2-like” thickness,
respectively, and their comparisons are now observed in the two panels at the
bottom of Fig. 7. According to our analysis, ICESat-like thickness tends to
underestimate the ice thickness by about 47.9 cm when MW99 is used in
comparison to OIB thickness, and CS2-like ice thickness shows an overestimate
of about 25.5 cm. Nevertheless, their correlation coefficients and RMSEs are
similar to the results obtained from the

The better agreement of

We have demonstrated that the method of simultaneously retrieving the sea
ice thickness and snow depth was successfully implemented with OIB
measurements. Now we extend the proposed approach to satellite freeboard
measurements. Here the method is tested with CS2 freeboard measurements,
solving for

Geographical distributions of observed CS2 radar freeboard
(

Monthly means of CS2-estimated freeboard (

Retrieved ice thickness from the CS2 freeboard (

To assess the accuracy of CS2 retrievals, reference snow depth and ice thickness collocated with CS2 freeboard in space and time are necessary. However, different from simultaneous retrievals from OIB freeboards in Sect. 4.2, the evaluation with the required matching data may not be possible from the monthly composite of CS2 data used in this study. Here, instead of using monthly collocated match-up data, an indirect way is used to examine the accuracy of CS2 retrievals. We do so by examining whether the relationship between the simultaneous method and the MW99 method, based on retrievals from the OIB freeboard, can be reproduced by CS2-based retrievals. If similar results are obtained, respective accuracies can be deduced against those noted from the evaluation against OIB measurements.

Comparison of simultaneously retrieved snow depth and ice thickness
to those from the MW99 method.

The relationships which can be obtained from the analysis in Sect. 4.2 – i.e.,

A new approach towards simultaneously estimating snow depth and ice
thickness from space-borne freeboard measurements was proposed and tested
using OIB data and CS2 freeboard measurements. In developing the algorithm,
the vertical temperature slopes were assumed to be linear within the snow
and ice layers so that a continuous heat flux could be maintained in both
layers. This assumption allowed for the description of the snow–ice
vertical thermal structure with snow skin temperature, snow–ice interface
temperature, the water temperature at the ice–water interface, snow depth,
and ice thickness. Based on the continuous heat transfer assumption, the
snow–ice thickness ratio (

Before applying the

In the retrieval process, we may be concerned about the applicability of the
algorithm developed with buoy observations representing the point
measurements to the larger spatial and temporal scales of satellite
measurements. This concern may be relevant upon observing the range of

Overall, the developed

The relationship found between

Distribution of physical variables on scatterplots of the
temperature difference ratio of snow and ice layer (

Histogram of estimated

First, the averages of basic properties available from buoy measurements are
compared. They include ice thickness, snow depth, snow–ice interface
temperature, ice temperature –

The thermal conductivities,

The calculated thermal conductivities are presented in Fig. A2. The
calculated

As a significant difference is observed in

Based on this knowledge, we can infer the condition of the snow layer of the
two parts. The relatively higher and varying

In summary, it is concluded that the physical properties of snow and ice can
account for the piecewise linearity based on the differences in the
physical properties between the first and second parts. Especially the
thermal conductivity of the snow,

Here we present results of a sensitivity test for showing how the snow depth
and ice thickness retrieval results are dependent on the uncertainties in

First,

With

It is also of importance to ask what degree of retrievals was yielded successfully. In this study, cases showing

The physical state of typical cases of points A, B, and C.

Errors of snow depth (

Locations of physical states for typical types (A, B, C) on the
freeboard-thickness ratio space. Blue dots are from

Statistics of success/fail ratios of

Although the sensitivity test regarding uncertainty of satellite-derived

Using Eqs. (C1) and (C2), uncertainties of snow depth and ice thickness
retrievals can be estimated. Ice thickness uncertainty estimates are
presented in Fig. C1. Total uncertainty of ice thickness estimates ranges
from 0.8 to 2.0 m. Generally,

Values and uncertainties of input variables for uncertainty estimation.

Geographical distributions of sea ice thickness uncertainty:
(first row) total uncertainty, (second row)

Geographical distributions of snow depth uncertainty: (first row)
total uncertainty, (second row)

The SHEBA buoy data were obtained from NCAR/EOL
(

The supplement related to this article is available online at:

HS and BJS conceptualized and developed the methodology, and HS conducted data analysis and visualization. GD, RTT, and SML gave important feedback for the algorithm development and interpretation of results. GD provided AASTI data. All of the authors participated in writing the paper; HS prepared the original draft under the supervision of BJS and GD, and BJS critically revised the paper.

The authors declare that they have no conflict of interest.

We appreciate NSIDC for producing and providing the OIB, CS2, and SIC datasets. We also give thanks to CRREL and NCAR/EOL under the sponsorship of the National Science Foundation for providing IMB and SHEBA buoy data. The authors express their sincere thanks to an anonymous reviewer and to Isobel R. Lawrence for their valuable comments that led to the improvement of the paper.

This study has been supported by the Space Core Technology Development Program (NRF-25 2018M1A3A3A02065661) of the National Research Foundation of Korea and by Korea Meteorological Administration Research and Development Program under grant KMIPA KMI2018-06910. This study has also been supported by the International Network Program of the Ministry of Higher Education and Science, Denmark (grant ref. no. 8073-00079B).

This paper was edited by Yevgeny Aksenov and reviewed by Isobel Lawrence and one anonymous referee.